By definition, for any $\bm{v}\equiv(\bm{\omega},\bar{\bm{\xi}}^{+})$,

(9.126)

$$\mathscr{N}(\bm{v})\equiv\mathfrak{F}_{0}\Big(\TF(-Q_{\beta})\Big)-\mathfrak{F}_{0}\Big(\TF(-Q_{\beta}-S_{d^{-}\bm{\eta}})\Big).$$

and hence

(9.127)

$$\mathscr{N}(\bm{v}_{1})-\mathscr{N}(\bm{v}_{2})=\mathfrak{F}_{0}\Big(\TF(-Q_{\beta}-S_{d^{-}\bm{\eta}_{2}})\Big)-\mathfrak{F}_{0}\Big(\TF(-Q_{\beta}-S_{d^{-}\bm{\eta}_{1}})\Big).$$

Since $\mathfrak{F}_{0}:\mathscr{S}_{0}(\mathbb{R}^{3})\to\mathscr{S}_{0}(\mathbb{R}^{3})$ is a smooth map on the space of trace-free symmetric $(3\times 3)$-matrices, there is some universal constant $C>0$ such that

(9.128)

$\displaystyle\begin{split}|\mathscr{N}(\bm{v}_{1})-\mathscr{N}(\bm{v}_{2})|&\leq C|d^{-}\bm{\eta}_{1}*d^{-}\bm{\eta}_{1}-d^{-}\bm{\eta}_{2}*d^{-}\bm{\eta}_{2}|\\ &\leq C(|d^{-}\bm{\eta}_{1}|+|d^{-}\bm{\eta}_{2}|)\cdot|d^{-}(\bm{\eta}_{1}-\bm{\eta}_{2})|.\end{split}$

Multiplying by the weight function,

(9.129)

$\displaystyle\begin{split}\rho_{\delta,\nu+1,\mu}^{(0)}(x)\cdot|\mathscr{N}(v_{1})&-\mathscr{N}(v_{2})|\leq C\cdot\rho_{\delta,\nu+1,\mu}^{(0)}(x)\cdot(|d^{-}\bm{\eta}_{1}|+|d^{-}\bm{\eta}_{2}|)\cdot|d^{-}(\bm{\eta}_{1}-\bm{\eta}_{2})|\\ &\leq C\Big(\rho_{\delta,\nu,\mu}^{(1)}(x)\cdot(|d^{-}\bm{\eta}_{1}|+|d^{-}\bm{\eta}_{2}|)\Big)\cdot\Big(\rho_{\delta,\nu,\mu}^{(1)}(x)\cdot|d^{-}(\bm{\eta}_{1}-\bm{\eta}_{2})|\Big).\end{split}$

Taking sup norms,

(9.130)

$\displaystyle\|\mathscr{N}(\bm{v}_{1})-\mathscr{N}(\bm{v}_{2})\|_{C^{0}_{\delta,\nu+1,\mu}(\mathcal{M})}\leq C\Big(\|\bm{v}_{1}\|_{C^{1}_{\delta,\nu,\mu}(\mathcal{M})}+\|\bm{v}_{2}\|_{C^{1}_{\delta,\nu,\mu}(\mathcal{M})}\Big)\cdot\Big(\|\bm{v}_{1}-\bm{v}_{2}\|_{C^{1}_{\delta,\nu,\mu}(\mathcal{M})}\Big).$

By similar computations, we also have the estimate for the Hölder seminorm

(9.131)

$\displaystyle\Big[\mathscr{N}(\bm{v}_{1})-\mathscr{N}(\bm{v}_{2})\Big]_{C^{0,\alpha}_{\delta,\nu+1,\mu}(\mathcal{M})}\leq C\Big(\|\bm{v}_{1}\|_{C^{1,\alpha}_{\delta,\nu,\mu}(\mathcal{M})}+\|\bm{v}_{2}\|_{C^{1,\alpha}_{\delta,\nu,\mu}(\mathcal{M})}\Big)\cdot\Big(\|\bm{v}_{1}-\bm{v}_{2}\|_{C^{1,\alpha}_{\delta,\nu,\mu}(\mathcal{M})}\Big).$

So we obtain the effective estimate (9.125) for the nonlinear errors. ∎